A) Are momentum, angular momentum and energy conserved? Under which circumstances?
B) Angular velocity of the ball immediately before the collision?
C) M/m rate so as to make the system (rod+ball) stay at rest after collision.
D) Velocity of the block after the collision.
E) Obtain an equation for D taking into account there is friction (Mu coefficient) ground-block along distance d and height h.

These are my answers:

A) - Linear momentum is conserved before/after the collision, due to the lack of external forces. However, there is a force exerted by the pivot on the system during the collision. Therefore, linear momentum of the system (ball+rod) IS NOT CONSERVED.

- Angular momentum is conserved due to the fact that torque exerted on the rod by the ball is equal and opposite to the torque exerted on the ball by the rod. (Meaning that AM of the rod is equal and opposite to AM of the ball; therefore AM of the system IS CONSERVED).
- Energy IS CONSERVED IF FRICTION of any kind IS NEGLECTED. If we considered friction ground-block along distance d, energy would not be conserved after the elastic collision.

B) First, I obtain system's center of mass: I consider the rod a puntual mass and remembering both rod and ball masses are equal to m: Xcm= (m(l/2)+ml)/2m= 3/4l
As energy is conserved before the collision and system start and ends at rest, we can assure gravitational potential energy transforms to kinetic energy(note omega equals to angular velocity):
2mg(3/4l)=1/2(2m(3/4l)^2)omega^2 Therefore: omega= sqroot(8g/3l)

C) First we obtain CoM's velocity before the collision:
V=(omega)radius; therefore Vcm:sqroot(3gl/2)
Now to obtain the velocity of the block after the collision we use momentum conservation principle:
2mVcm=MVblock; therefore: Vblock=2m/Msqroot(3gl/2)
Finally, to relate both M and m we use the principle of conservation of kinetic energy at an elastic collision:
1/22mVcm=1/2MVblock; therefore: M/m=2

D) Vblock= sqroot(3gl/2)

E) Energy is not conserved along d, however we now deltaE=muMgd. Therefore, Vblock= (mu)gd.
We have a projectile trajectory. Block has an initial velocity just at its horizontal component. Therefore:
D= d + Vblock(t) and y= h - 1/2g(t^2) Solving for t at second equation: t=sqroot(2h/g)
Finally, D= d(1 + mu(g)sqroot(2h/g))

Please if you could check my answers I would be greatful. If there's anything wrong or that can be improved let me know.
Thanks

I tried to post the pic as you said but did not work.
The text says the following: there is a system (ball+rod) which collides elastically against a block. The system starts at rest, besides, after the collision the system stays at rest too. The block is at rest until is hit by the system. The block goes along a distance d before falling from a height h, reaching a distance D. There's friction between block and ground. To sum up, I want to point out that even if I'm right with the answers if you wanted to add something in order to improve them, it would be great. Thanks

1 Answer

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Best answer

A) Your answer is not clear about whether linear momentum is conserved. As you state, there is an external force acting during the collision between the ball $m$ and the block $M$. This external force is the reaction at the hinge (pivot), which is attached to the outside world. This means that linear momentum might not be conserved. However, angular momentum about the hinge is conserved.

B) The question states that "the system" is at rest after the collision. Doesn't this mean the ball + rod? You have calculated the angular velocity immediately before the collision.

I can see that you have equated PE lost to KE gained, using $\frac12 I \omega^2$ for KE. But how have you calculated moment of inertia $I$?

The moment of inertia of the rod about one end is $\frac13 mL^2$, not $m(\frac12 L)^2=\frac14 mL^2$. The moment of the (point) mass $m$ is $mL^2$. So the total moment of inertia of ball+rod is $\frac43 mL^2$ not $\frac98 mL^2$.

Your calculation method for moment of inertia is wrong. The moment inertia of an extended object is not the moment of inertia of its CM. For example, if the rod were pivoted at its CM, your method gives MI=0 instead of $\frac{1}{12}mL^2$. However, the method does work for calculating the change in PE (provided the gravitational field is uniform).

C) Here you need to apply conservation of angular momentum not linear momentum.

The KE of the ball+rod immediately before the collision equals the KE of the block immediately after the collision :
$\frac12 I\omega^2=\frac12 Mu^2$
Also, the angular momentum of the ball+rod equals that of the block :
$I\omega=MuL$
so
$I^2\omega^2=M^2u^2L^2$
Dividing the AM equation by the KE equation we get
$I=ML^2$
However, $I=\frac43 mL^2$ therefore $M=\frac43 m$.

D) The velocity $u$ of the block immediately after the collision is given by $\frac12 Mu^2=32mgL$ where $M=\frac43m$ therefore $u=\frac32\sqrt{gL}$.

E) Your calculation of the velocity of the block as it leaves the table does not seem to be correct.

The initial KE of the block before it slides across the rough part of the table is $\frac12 Mu^2=\frac12 M \frac94 gL=\frac98 MgL$ from above. The work done on the block by friction is $\mu Mgd$. So the velocity $v$ of the block as it leaves the table is given by
$\frac12 Mv^2=\frac98 MgL-\mu Mgd$
$v=\sqrt{(\frac94L-2\mu d)g}$.
It is not possible to simplify this expression unless you are given a value for $\mu d$ in terms of $L$. EG if $\mu d=\frac58 L$ then $v=\sqrt{gL}$.

Judging by the diagram $D$ is measured from the edge of the table, not from the initial position of the block. Then $D=vt$ where $h=\frac12 gt^2$ which you have got right.

Yes, I calculated the angular velocity immediately before the collision.
We know moment of inertia equals to mr^2. Therefore: I=2m(3/4l)^2. I know r=3/4l Thanks to the fact I calculated the system's Center of Mass

The moment of inertia of the rod about one end is $\frac13 mL^2$, not $m(\frac12 L)^2=\frac14 mL^2$. The moment of the (point) mass $m$ is $mL^2$. So the total moment of inertia of ball+rod is $\frac43 mL^2$ not $\frac98 mL^2$.

Your calculation method is wrong. The moment inertia of an extended object is not the moment of inertia of its CM. For example, if the rod were pivoted at its CM, your method gives MI=0 instead of $\frac{1}{12}mL^2$.

Note I did not took 1/2L as the radius. What I did was first, calculate the Center of mass of the system (rod+ball) which equals to 3/4L. That's the value I considered as my 'height'. Therefore as initial potential energy of the system I do have 2mg(3/4L) . Note I wrote down 2m because the system is composed by ball+rod of equal masses. Until this point do you agree with me?

Okey, now I would like to explain you what I did at C. Firstly I calculated the velocity of the center of mass of the system before its collision. As we know: v=omega*radius. Therefore: Vcm=9/8*L*sqroot(g/L)=9/8*sqroot(gL)
Once I got Vcm, as we know linear momentum is conserved before and after the collision we can assure the following: 2mVcm=MVblock.
Therefore: Vblock=(2*m*9*sqroot(g*L))/(8M)
As we know the collision is elastic, kinetic energy is conserved. Therefore:
1/2*2*m(9/8*sqroot(g*L)^2 = 1/2*M*(2m/M*9/8*sqroot(g*L))^2
Therefore M/m=2

Linear momentum is not conserved during the collision, as you acknowledged in your answer to part A. There is an external force (at the pivot) which acts during the collision. Angular momentum is conserved.

Linear momentum is not conserved during the collision, we both agree with that. But note linear momentum IS conserved before and after the collision, due to the lack of external forces. Therefore what I did should be right, as I considered momentum before the collision (2mVcm) to be equal to momentum after the collision (MVblock). Do you agree? As far as I'm concerned we both are right

No we do not agree. Linear momentum of the ball+rod before the collision is not equal to linear momentum of the block after the collision. An external force acts during the collision, ie between "before" and "after".

If conservation of linear momentum gives a different answer to conservation of angular momentum, then we cannot both be right.

Okey, I'm trying to obtain the rate M/m using conservation of angular momentum and conservation of kinetic energy (elastic collision), but I'm struggling to get it. I'm using the following equations:
1) CONSERVATION OF AM: 2m*(3/4L)*9/8sqroot(gL)=I*omega; I=4/3m(L^2)+M(L^2) and omega=3/2sqroot(g/L)
2) CONSERVATION OF KE: 1/2*(2m)*(9/8sqroot(gL)^2)=1/2MVblock.
The problem is that I'm not getting a clear result for M/m
And Vblock should be 9/8sqroot(gL), as the velocity of the block is the one corresponding to the velocity of the system (we negligited friction until section E)

Thanks I realized what I did wrong! I got the velocity of the block (asked to calculate it at D)) as follows:
As the block is at rest before the collision, Therefore Vblock should be equal to Vcm of the system, which equals to 0 after the collision. Then:
Vcm=omega*radius=9/8sqroot(gL)=Vblock
What do you think about it?

I get your point, you applied conservation of energy of the system. But why am I wrong? Shouldn't be Vcm of the system before the collision equal to Vblock? Why not then? You got as a result Vblock equals to angular velocity of the system before the collision, which has a lot of sense.

Then to sum up, I want to expose what I did at E)
I got D = Vblockt therefore D= 3/2sqroot(2hL)
Do you agree with this result?

The impulse at the pivot means that linear momentum is not conserved, but angular momentum is conserved. Therefore you cannot use the motion of the CM at impact. Also, if two objects of equal mass collide elastically then their velocities are switched, but in this case the colliding objects have unequal masses.

To avoid confusion I use $u=\frac32\sqrt{gL}$ as the velocity of the block immediately after the collision. This is different from the velocity $v$ with which it leaves the table, after it has moved the distance $d$ and lost some KE due to friction.

I have updated my answer to provide an expression for $v$, which should include $d, \mu$ unless you have been given values for these. $D=vt$ is correct but you are using the wrong value for $v$.

The block loses kinetic energy because of friction. If $d$ were infinite the the block would have to start with an infinite amount of kinetic energy in order to get to the edge of the table. If the block starts with a finite amount of kinetic energy, it can only go a finite distance before losing all that energy and stopping.

In this case the block stops (ie $v \to 0$) when $\frac94L = 2\mu d$. This gives you the maximum distance which the block can travel. $d$ cannot increase any further than this, so $v$ does not become a complex number.